Advancements in Anisotropic Littlewood-Paley Theory and Global Regularity of Navier-Stokes Equations

This paper presents significant advancements in the anisotropic Littlewood-Paley theory and its application to the global regularity framework of the Navier-Stokes equations. By extending the classical decomposition to non-Euclidean geometries, particularly on Riemannian manifolds, we provide a robust theoretical foundation for analyzing fluid dynamics in complex domains. Fractional Sobolev spaces are employed to refine regularity criteria, addressing fine-scale structures and anisotropic behaviors in turbulent flows. The integration of quaternionic structures introduces a novel perspective for capturing rotational symmetries, allowing for a deeper understanding of bifurcations in three-dimensional fluid systems. These structures are leveraged to establish new theorems on the existence and uniqueness of solutions, with precise energy estimates in Besov spaces. Furthermore, we rigorously analyze the dissipation of energy across scales, demonstrating how high-frequency components dominate the cascade process in turbulent regimes. Key results include a detailed formulation of anisotropic Sobolev embeddings, explicit regularity theorems for weak solutions in fractional spaces, and enhanced energy dissipation models derived through advanced harmonic analysis techniques. The paper also addresses the stability and bifurcation of solutions within a quaternionic framework, offering insights into the interplay between geometry, nonlinearity, and dissipation. Overall, this work lays the groundwork for addressing open problems in fluid dynamics, including the Millennium Prize Problem on the Navier-Stokes equations, and sets a path for future explorations in anisotropic and geometric fluid mechanics. The results are supported by rigorous proofs and detailed mathematical frameworks, ensuring broad applicability to theoretical and computational studies.